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Simple Harmonic Motion and the Simple Pendulum
PHYS 1313
Prof. T.E. Coan
version: 30 Jul ‘99
Introduction
It is a common day experience to see objects oscillating back and forth. We have, for
example, all seen tree branches sway back and forth in the wind. What we may not have
noticed is that for many oscillating objects, the time for the object to oscillate away from
its original position and then to return to this same position, does not depend on the size
of the oscillation. Small oscillations take as long as large oscillations. Technically, we say
that the “period” of oscillation, the round trip time if you will, is independent of the
displacement. This phenomenon is called simple harmonic motion. It is actually quite
common in nature. What is not immediately obvious is that objects subject to a force that
can be expressed in a form similar to Hooke’s law execute simple harmonic motion.
Simplified Theory
Another simple device besides the spring-mass that can produce simple harmonic motion
is the simple pendulum. A simple pendulum consists merely of a point-mass (m)
suspended from a fixed point by a rod or string of length (L). The mass of the rod or
string is assumed to be so much less than the suspended mass that it can be ignored. If the
suspended mass is displaced to the left or right, while the rod or string is kept taut, and
then released, the mass will swing freely back and forth under the gravity’s influence. For
small horizontal displacements (angular displacements of less that 10°, for instance), the
restoring force on the suspended is given by:
F = ma = m (- g/L) x
(1)
Here, m is the mass of the point-mass or bob, g is the magnitude of the gravitational
acceleration, L is the length of the string or rod, and x is the horizontal displacement. The
presence of the minus sign is not a typographical error and merely says that the restoring
force points in a direction opposite to the displacement. Since m, g, and L are positive
constants, we note that formula (1) has the same form as the equation for Hooke’s Law.
Recall that Hooke’s law is of the form (F = -k x ). Although we won’t prove it, it can be
shown that an object is subject to Hooke’s law, or any other force that can be expressed
in a form similar to Hooke’s law, it will execute simple harmonic motion. We therefore
expect that the suspended mass of the simple pendulum will execute simple harmonic
motion.
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Since real pendulums have strings or rods that certainly aren’t massless, you might ask
when is it reasonable to treat a real pendulum as an ideal pendulum? A real pendulum
swinging with small amplitude may be described as a simple pendulum if:
1. The mass of the string or rod supporting the mass is much less than the mass of the
bob.
2. The length of the string or rod (L) is much greater than the size of the bob. In this
case, the suspended mass can be treated as a point-mass located at its center of mass.
One of the easiest things to observe about a simple pendulum executing simple harmonic
motion is its period of oscillation. Theoretically, the period is given by:
T = 2  L/g
(2)
Notice that the mass of the suspended bob does not appear. This implies that mass of the
bob does not influence the period of the oscillation as long as the above two conditions
are met. So, two pendulums of the same length, standing side by side but with bobs of
very different masses, will have identical periods.
Squaring both sides of Equation (2) yields:
L / T2 = constant
(3)
This equation says that, for simple harmonic motion, the ratio of the length of the
pendulum length to the square of its period is a constant. This means that if the pendulum
length is increased, the period will also increase in such a way that equation (3) is
satisfied.
Furthermore, solving for g yields
g = 4 2 L/T2
(4)
Our goal is to observe simple harmonic motion for a simple pendulum. We will take
measurements of L and T for a real pendulum and then can use (3) to test the validity of
the theory (L / T2 = const.). Furthermore, by using Equation (4) we can determine an
experimental value for g, the acceleration due to the Earth’s gravity for objects near the
Earth’s surface.
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Procedure
1. Clamp one end of a piece of thread to a rigid support in such a way that insures that
the point of suspension remains fixed throughout a complete (but small-angle) cycle.
2. Attach the bob to the other end of the thread. Beginning with a pendulum length of
slightly less than 100 cm. (from the point of suspension to the center of the bob), start
the bob swinging through a small arc ( about 5 ° on either side of vertical).
3. Use the timer to find the time for about 20 cycles at this L. Note that the timer should
be started at the zero count and stopped at the nth count. (n is the number of cycles.)
Record L, t, and n.
4. Change L to about 60 cm. Find the time for about 25 cycles. Record L, t, and n for
this new arrangement.
5. Change L to about 30 cm. Find the time for about 30 cycles. Record L, t, and n for
this new arrangement.
Analysis
For each trial, compute the theoretical and experimental period. The theoretical period
can be obtained from Equation (2), while the experimental period can be obtained from
t
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n Compare these values. Compute the ratio L/T exp, where the period T is the
experimentally observed period. Record these values in the table. Use the experimental
period and Equation (4) to compute g and record this in the table provided.
Conclusions
1. Describe your results for this experiment.
2. Why was it desirable to use a greater n for smaller L values
3. Do you think the string and the hanging mass approximates a simple pendulum?
Defend your position.
4. If you have a pendulum clock that runs too slow, should you lengthen or shorten the
pendulum to correct it?
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Error Analysis
1. How well do you think you can measure the pendulum length? Explain.
2. How well do you think you can measure the period of oscillation? Explain.
3. Compare your value for g to the accepted value. How do they compare? Compute the
percentage difference between your answer and the accepted value. If the difference
seems large, what do you think caused this difference?
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The Simple Pendulum
PHYS 1313
Prof. T.E. Coan
Version: 30 Jul ‘99
Name _________________________Section ________________
Abstract:
Data:
L
t
n
Ttheoretical Texperimental L/T2exp
gave
Calculations:
=_________
g
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Conclusions:
Error Analysis: